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Investigate the Mathematics Behind the Tuning Systems of Wendy Carlos (15)

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Investigate the mathematics behind the tuning systems of Wendy Carlos (15) Student number: 200666371 Thursday 10th November 2011 Introduction - The twelve-tone equal temperament has been the dominant tun- ing system used in Western music for centuries. However, this has not always been the case. There exists a large variety of different tunings, more or less used depending on time and place, from the Pythagorean scale to meantone tempera- ment, or just intonation. Today, computers and synthesizers allow musicians to invent even more tunings because there are no material restrictions anymore: it is not necessary to construct a new instrument in order to use a new scale. Thus, electronic music is very well adapted to experiments in musical tuning. Wendy Carlos is an electronic musician and composer very interested in exotic sounds and alternative tunings. She is famous for having composed the original soundtrack of Kubrick's film A Clockwork Orange. She experimented several alternative scales, in particular in her album The Beauty in the Beast. Some of these scales were her own creations: three equal-tempered scales called alpha, beta and gamma, and a \super-just" scale called the harmonic scale. We will explain on which math- ematical basis are constructed these scales, and in what extent this makes them sound good. All musical scales try to approximate simple integer ratios for the frequency ra- tios, in order to have the most consonant tuning. In equal-tempered scales, the frequency ratio of two consecutive notes is constant, so the degree (distance be- tween consecutive pitches) should be chosen in order to get as close as possible to these ratios. Most existing equal-tempered tunings are based on an interval with frequency ratio 2:1, the octave, which is divided into an equal number of steps: 12, 19, 24, 31, etc. 1 The alpha, beta and gamma scales are equal-tempered scales, but their par- ticularity comes from the fact that the octave is not divided by a whole number. Ignoring the octave, equal temperament can obtain better approximations for the other simple integer frequency ratios, such as the perfect fifth 3:2 or the major third 5:4. In [4], Carlos said that she \discovered" these three scales while trying 3 5 6 7 11 to find good approximations for the ratios 2 , 4 , 5 , 4 , and 8 . A method on how to find frequency ratios close to these simple integer ratios is explained in [1]. Here, we will only consider the perfect fifth (ratio 3:2) and the major third (ratio 5:4), but we will actually find also a good approximation for the minor third (ratio 6 3 5 6:5) as 5 = 2 ÷ 4 . We need to find a scale degree x for which there exist some integers m and n such that the perfect fifth occur at approximately m steps, and the major third at approximately n steps, that is: 3 5 n × x × log ≈ m × x × log 2 2 2 4 3 5 In other words, the ratio log2 2 ÷ log2 4 should approximate an integer ratio, and small integers would be better, in order not to get too much notes within an octave. The best way to find a rational approximation of this ratio is using continued fractions: 3 log2 2 1 5 = 1 + 1 log2 1 + 1 4 4+ 1 2+ 1 6+ 1+::: 1 9 The alpha scale is based on the approximation 1 + 1 = 5 for this ratio. So with 1+ 4 our previous notation, we have that m = 9 and n = 5, and as a consequence, 4 steps approximate the 6:5 minor third. In order to find the best x for our scale, we have to minimise the mean square deviation: this is a function describing the difference between the model we want to follow and the reality. This corresponds to: 3 5 6 (9x − log )2 + (5x − log )2 + (4x − log )2 2 2 2 4 2 5 (where the unit is the octave). The minimum value for this function is obtained when the derivative with respect to x equals zero, that is: 9 log 3 + 5 log 5 + 4 log 6 x = 2 2 2 4 2 5 ≈ 0:06497 92 + 52 + 42 and we find x = 0.06497... octave. A better unit to express this value is the cent, where the octave is divided into 1200 cents. So this scale is of degree 78 cents, and there are approximately 15.4 steps per octave. The steps for beta and gamma scales are found with the same calculations, 3 5 but with different approximations for the ratio log2 2 ÷ log2 4 . For beta we choose 2 1 11 1 20 1 + 1 = 6 found by rounding up. For gamma, we take 1 + 1 = 11 . We 1+ 5 1+ 1 4+ 2 find that beta has 18.8 steps to the octave, and gamma 34.2 (respectively 63.8 and 35.1 cents per step). Figure 1: α, β and γ scales compared to exact ratios, from [1] As you can see, these scales have better approximations for the simple ratios than the 12-tone equal-tempered scale, where there are 300 cents for the minor third, 400 cents for the major third and 700 cents for the fifth. Alpha and beta have very similar properties, but the sevenths are a little more in tune in beta. Gamma is closer to just tuning for minor thirds, major thirds and fifths, which are almost perfect; this scale wasn't used on the album The Beauty in the Beast. As we said earlier, consonance occurs when frequencies of notes are related by simple integer ratios: two pitches with a small integer ratio between their frequencies are in perfect harmony. A possible tuning would be to construct notes with frequencies following exactly these ratios. Such tuning systems are called just intonation. The main problem of these scales is that all the pitches are constructed in relation to a single note, so they are in harmony while we play in that key, but we can't modulate (meaning start to play in another key) because some pitches will be dissonant with the new tonic. So with the classic 12 notes per octave, we can play in one key only. Wendy Carlos invented a 'super-just' scale in the sense that it extends just intonation, beyond 5-limit. In other words, the exact ratios of the frequencies use multiples of primes other than 2, 3 and 5. This is the harmonic scale. She described this in [2] and in [3]. This scale has 144 different pitches per octave. This corresponds to 12 keys × 12 notes in a chromatic scale. The principle is to consider a \root note" or key, and to construct a just scale related to this note. This scale will be composed of harmonics of a low fundamental, which means notes with a multiple of its frequency. For more clarity, let's construct this scale with the root C for example. The 12 notes of the octave related to C are harmonics of 3 a low C, called the fundamental. Their wavelength ratio (from the fundamental) are terms of the harmonic series. Figure 2: Harmonics of C, from [2] The first C in our octave will be the 16th harmonic of this fundamental. Then, C] is the 17th harmonic, D the 18th, E[ the 19th, E the 20th, F the 21st, F] the 22nd, G the 24th, A[ the 26th, A the 27th, B[ the 28th and B the 30th. Finally, C is the 32nd harmonic. We can then work out the frequency ratios of each note of the octave from C, and they are simple integer ratios: Figure 3: Harmonic scale on C table, from [2] You can notice that E is the perfect major third, G is the perfect fifth. All the other ratios are simple too, so all the notes will sound good with C. But if we use only this table, transpositions from one key to another are not possible. Indeed, 16 13 26 in the key of D[, the fifth would be A[, but then the ratio is 17 × 8 = 17 , which is not perfect anymore. That is why, for her harmonic scale, Carlos established such tuning tables for every note in the 12-tone equal-tempered scale: each one plays the role of the root, and we will thus obtain a 12-tone harmonic scale for each of them, hence the 144 pitches per octave. A harmonic scale for a certain root is away from the previous one by 100 cents, that is one step in 12-tone equal temperament. With all these pitches, we can now modulate between the distinct keys, and the just intonation restriction of playing in only one key disappears. 4 While playing music with the harmonic scale, these transpositions are controlled by the appropriate note on a one-octave music keyboard, which automatically retunes all the notes of the playing keyboard according to the corresponding tuning table. One interesting phenomenon of this scale is that if you play several notes of one tuning table, you can ear a low note: the fundamental. As the frequencies of the notes are multiples of the fundamental, when we add them we obtain a periodic wave, with the same period as the fundamental wave. Conclusion - The mathematics permit to invent new tuning systems, more con- sonant, and electronic music provide the technical tools to put them into practice. It is of course easier to tune instruments in 12-tone equal-tempered scale than in Carlos' scales, because in alpha, beta and gamma there is no octave, and in the harmonic scale we have 144 notes per octave! But when music is played using synthesizers, there is no reason to keep this scale as a standard anymore.
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